Simplify $\frac{\log \sqrt{27}}{\log \sqrt{81}}$

Analyze the question: We need to simplify a ratio of two logarithms. Note that this is not log(27/81). It is a division of two separate logs, so we cannot use the quotient law directly. Instead, we rewrite each log using powers of 3.

Concept:

• √x = x^(1/2)
• log(aⁿ) = n log a

Solution:

Numerator: √27 = (3³)^(1/2) = 3^(3/2) so log √27 = (3/2) log 3

Denominator: √81 = (3⁴)^(1/2) = 3² so log √81 = 2 log 3

Now divide:

Answer: D. 3/4.

Tip: When you see a ratio of logs of powers of the same base number, convert each to a multiple of the same log (here log 3). The log 3 cancels, leaving a simple fraction. Also remember: log a / log b is NOT log(a/b).